/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 21 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
   active(f(X1, X2)) -> f(active(X1), X2)
   active(g(X)) -> g(active(X))
   f(mark(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(g(X)) -> g(proper(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))

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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(mark(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(mark(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
mark0(0) -> 0
ok0(0) -> 0
proper0(0) -> 0
active0(0) -> 0
f0(0, 0) -> 1
g0(0) -> 2
top0(0) -> 3
f1(0, 0) -> 4
mark1(4) -> 1
g1(0) -> 5
mark1(5) -> 2
f1(0, 0) -> 6
ok1(6) -> 1
g1(0) -> 7
ok1(7) -> 2
proper1(0) -> 8
top1(8) -> 3
active1(0) -> 9
top1(9) -> 3
mark1(4) -> 4
mark1(4) -> 6
mark1(5) -> 5
mark1(5) -> 7
ok1(6) -> 4
ok1(6) -> 6
ok1(7) -> 5
ok1(7) -> 7

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(6)
BOUNDS(1, n^1)

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(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
   active(f(X1, X2)) -> f(active(X1), X2)
   active(g(X)) -> g(active(X))
   f(mark(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(g(X)) -> g(proper(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(9) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

f(mark(X1), X2) ->^+ mark(f(X1, X2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X1 / mark(X1)].

The result substitution is [ ].




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(10)
Complex Obligation (BEST)

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(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
   active(f(X1, X2)) -> f(active(X1), X2)
   active(g(X)) -> g(active(X))
   f(mark(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(g(X)) -> g(proper(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
   active(f(X1, X2)) -> f(active(X1), X2)
   active(g(X)) -> g(active(X))
   f(mark(X1), X2) -> mark(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(g(X)) -> g(proper(X))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL